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this right here is a picture of Rene Descartes once again one of the great minds in both math and philosophy and I think you'll you're seeing a little bit of a trend here that the great philosophers were also great mathematicians and vice versa and he was somewhat of a contemporary of Galileo he's 32 years younger although he died he died shortly after Galileo died this guy died much at a much younger age Galileo was well into his 70s decart died at what this is only at 54 years old and he's probably most known in popular culture for this quote right over here a very philosophical quote I think therefore I am but I also wanted to throw in and this isn't that related to algebra but I just thought it was a really neat quote probably his least famous quote this one right over here and I like it just because it's very practical it makes you realize that these great minds these pillars of philosophy and mathematics at the end of the day they really were just human beings and he said you just keep pushing you just keep pushing I made every mistake that could be made but I just kept pushing which i think is very very good life advice now he did many things in philosophy and mathematics but the reason why I'm including here as we build our foundations of algebra is that he is the individual most responsible for a very strong connection between algebra and geometry so on the left over here you have the world of algebra and we've discussed it a little bit you have equations that deal with symbols and these symbols are essentially they can take on values so you could have something like y is equal to 2x Y is equal to 2x minus 1 this gives us a relationship between whatever X is and whatever Y is and we can even set up a table here and pick values for X and see what the values of Y would be and I could just pick a random valid random values for X and then figure out what Y is but I'll pick relatively straightforward values and just so that the math doesn't get too complicated so for example if X is negative 2 then Y is going to be 2 times negative 2 minus 1 2 times negative 2 minus 1 which is negative 4 negative 4 minus 1 which is negative 5 if X is negative one then Y is going to be 2 times negative 1 2 times negative 1 minus 1 which is equal to this is negative 2 minus 1 which is negative 3 if X is equal to 0 X is equal to 0 then Y is going to be 2 times 0 minus 1 2 times 0 0 minus 1 is just negative 1 I'll do a couple more if X is 1 and I could have picked any values here I could have said what happens if X is the negative square root of 2 or what happens if X is negative five-halves or positive positive 6 7 but I'm just picking these numbers because it makes the math a lot easier when I try to figure out what Y is going to be but when X is 1 Y is going to be 2 times 1 minus 1 2 times 1 is 2 minus 1 is 1 and I'll do one more I'll do one more and a color that I have not used yet let's see this purple if X is 2 then Y is going to be 2 times 2 now our X is 2 minus 1 minus 1 so that is 4 minus 1 is equal to 3 so fair enough I just kind of sampled this relationship I said ok this describes a general relationship between a variable Y and a variable X and then I just made a little bit more concrete I said ok well then if X is one of these variable for each of these values of X what would be the corresponding value of y and what Descartes realized is is that you could visualize this one you could visualize these individual points but that could also help you in general to visualize this relationship and so what he essentially did is he bridged the world of this kind of very abstract symbolic algebra and that end geometry which was concerned with shapes and sizes and angles so over here you have the world of you have the world of geometry and obviously there are people in history may be many people who history may have forgotten who might have dabbled in this but before Descartes it's generally viewed that geometry was Euclidean geometry and that's essentially the geometry that you study in a geometry class in in eighth or ninth grade or 10th grade in a traditional high school curriculum and that's the geometry of studying the relationships between and their angles and the relationships between circles and you have radii and and then you have triangles inscribed in circles and all the rest and we go into some depth in that in the geometry playlist but Descartes says well I think I can represent this visually the same way that Euclid was studying these triangles and these circles he said well why don't I if we view a if we view a piece of paper if we if we think about a two dimensional plane you could view a piece of paper as kind of a section of a two-dimensional plane we call it two dimensions because there's two directions that you could go in there's the up-down direction that's one direction so let me draw that I'll do it in blue because we're trying to do we're trying to visualize things so I'll do it in the geometry color so you have the up-down direction you have the up-down direction and you have the left/right direction that's why it's called a two-dimensional plane if we're dealing in three dimensions you would have an in/out dimension and out and it's very easy to do two dimensions on the screen because the screen is two-dimensional and he says well you know there are two variables here and they have this relationship so why don't I associate each of these variables with one of these dimensions over here and by convention let's make the Y variable which is really the dependent variable the way we did it it depends on what X is let's put that on the vertical axis and let's put our independent variable the one where I just randomly picked values for it to see what Y would become let's put that on the horizontal axis and it actually was Descartes who came up with a convention of using X's and Y's and we'll see later Z's in algebra so extensively as unknown variables or the variables that you're manipulating but he says well if we think about it this way if we number if we number these these these dimensions so let's say that in the X direction let's make this right over here this right over here is negative three let's make this negative two this is negative 1 this is zero now we're just i'm just number numbering the X direction the left/right direction now this is positive 1 this is positive 2 this is positive 3 now we can do the same in the Y direction so let's see we go so this could be let's say this is negative 5 negative 4 negative 3 negative actually let me do it a little bit neater than that let me clean this up a little bit so let me erase this and extend this down a little bit so I can go all the way down to negative five without making it look too messy so let's go all the way down here and so we can number it this is one this is two this is three and then this could be negative one negative two and these are all just conventions it could have been label the other ways we could have decided to put the X there and the Y there and make this the positive direction make this the negative direction but this is just the convention that people adopted starting with descartes negative two negative three negative four and negative five and he says well I think I can associate I can associate each of these pairs of values with a point on in two dimensions I can take the x-coordinate I can take the x value right over here I say okay that's negative two that would be right over there on the along the left right direction I'm going to the left because it's negative and that's associated with negative five in the vertical direction so I say the Y value is negative five and so if I go two to the left two to the left and five down and five down I get to this point right over there so he says the this these two values negative two negative five I can associate it with this point in the core in this plane right over here in this two-dimensional plane so I'll say that point is has the coordinates tells me where to find that point negative 2 negative 5 and these coordinates are called Cartesian coordinates named for Rene Descartes is he's the guy that came up with these he's associating all of a sudden these relationships with points on a coordinate plane and then he says well ok let's do another one there's this other relationship where I have when X is equal to negative 1 when X is equal to negative 1 Y is equal to negative 3 so X is negative 1 Y is negative 3 that's that point right over there and the convention is once again when you list the coordinates you list the x coordinate then the y coordinate and that's just what people decided to do negative 1 negative 3 that would be that point right over there and then you have the point when x is 0 Y is negative 1 when x is 0 right over here which means I don't go to the left or the right Y is negative 1 which means I go 1 down that's that point right over there zero negative one right over there and I could keep doing this when X is 1 Y is 1 when X is 1 Y is 1 when X is 2 y is 3 when X is 2 y is 3 actually we do them that same purple color when X is 2 y is 3 2 comma 3 and then this one right over here in orange was 1 comma 1 and this is neat by itself I essentially just sampled possible X's but what he realizes not only do sample these possible X's but if you just kept sampling X's if you've tried sampling all the X's in between you would actually end up plotting out a line so if you were to do every possible X you would end up getting a line that looks something like a line that looks something like that right over there and any any relation if you pick any X and and find any y it really represents a point on this line or another way to think about it any point on this line represents a solution to this equation right over here so if you have this point right over here which looks like it's about X is 1 and 1/2 Y is 2 so let me write that 1.5 1.5 comma 2 that is a solution to this equation when X is 1 point 5 2 times 1.5 is 3 minus 1 is 2 that is right over there so all of a sudden he was able to bridge this gap or or this this this relationship between algebra and geometry we can now visualize all of the all of the X and y pairs that satisfy this were this equation right over here and so he is responsible he is responsible for making this bridge and that's why the coordinates that we use to specify these points are called Cartesian coordinates and as we'll see the first type of equations we will study our equations of this form over here and in a traditional algebra curriculum they're called linear equations linear equations and you might be saying well you know ok this is an equation I see that this is equal to that but what's so linear about them what what makes them look like a line and to realize why they are linear you have to make this jump that Renee a cart made because if you were to plot this using Cartesian coordinates on a Euclidean plane you will get a line and in the future we'll see that there's other types of equations where you won't get a line well you'll get a curve or something something kind of crazy or funky